Very pretty! Looks like some sort of dwarf Azalea, but don't quote me on that.

Does your pot have drainage holes? That is a must - even with drainage, sometimes glazed pots hold too much water if you don't have bonsai mix with very good drainage. I also use a plastic mesh to cover the drainage holes, keeping mix from falling out the bottom. This mesh is also sold as needle point material. And you probably need metal wire to fasten the plant in the pot.

I would definitely get bonsai mix - even cacti soil usually holds too much water and doesn't have enough pockets for air. Repotme Gritty Mix Special Imperial is basically "Al's Gritty Mix" but somewhat more coarse and cleaner than the sellers on Etsy / eBay. It's expensive, but good and easier than sourcing the materials yourself.

There are really good videos on how to trim (and wire) a bonsai - you will eventually want to trim it to open up the middle, create some shape and let air and light flow through. However, for the time being, the only trimming I would do sooner than later is snipping off all those "suckers" at the bottom. Some basic trimming rules:

• Remove trifurcations: at most, you want a 2-way split at any one point, or node, on the plant. If 3 or more branches / trunks starts at or close to the same spot, you probably want to pick 2. This is for here-and-now aesthetic as well as to keep the plant growing healthy (it will bulge if there are too many branches at one spot).
• Remove upward and downward growing stems that come off of branches.
• Remove inward growing branches (branches that grow back in toward the center of the plant).
• Remove crossing branches: if two branches cross, you only need one.
• Think of the shape and style plant you want and shape it toward that goal over time: is, more of a bush? Single trunk? Multi-trunk? Banyan? Look at pics if the same species and see what you like, that also fits the material you have to work with (by material, I mean your plant).

I find if I keep these guidelines in mind, it makes a lot of decision making easier and the plant sort of shapes itself.

It is a variagated serissa and fairly easy to take care of. It can stand pretty cool temperatures but not freezing and so needs to be brought in. Takes pruning very well and Jon above has given you a good set of directions. It will flower all year round - light pink flowers in this case it looks like. It is also called 'snow rose' I believe. One peculiar aspect of this plant is that when you repot or root prune it will smell awful - its scientific name is serissa foetida for that reason.